Duke Physics Course PHY230 Outline
Mathematical Methods in Physics
Numbers in parentheses are sections in Arfken related to the material.
Topics and section numbers in square brackets are tentative; only some
will be covered.
Integration
Gaussian integrals.
Gamma function.
Beta function.
Volume of n-sphere.
Invariant integration.
Numerical quadrature.
Monte-Carlo integration.
Principal part integrals.
Miscellaneous functions defined by integrals (incomplete gamma, exponential
integrals, sine and cosine integrals, Fresnel integrals, error functions,
etc.).
Elliptic integrals.
[Elliptic functions].
(5.8, 10.1, 10.3-10.5)
Infinite Series
Convergence and convergence tests.
Zeta function.
Summing infinite series.
[Rational sums and polygamma functions].
Series of functions and uniform convergence.
Power series.
Series expansion methods.
Bernouilli numbers.
[Euler-Maclaurin formula].
Trigonometric Fourier series.
Convergence of Fourier series and Gibbs phenomenon.
Complex Fourier series.
Half-range Fourier series.
Asymptotic series.
(5.1-5.7, 5.9-5.10, [10.2], 14.1-14.6)
Functions of a Complex Variable
Analytic functions.
[Conformal mapping].
Contour integrals.
Cauchy's theorem and integral formula.
Taylor series.
Analytic continuation.
Laurent series.
Poles, zeroes, and branch points.
Residue theorem.
Contour integration.
Steepest descents.
(5.6, 6.1-6.6, 7.1-7.2, 7.4, [6.7 ])
Fourier Transforms
Delta functions and delta sequences.
Limit of Fourier series.
Definitions.
Convergence.
Examples.
Sine and cosine transforms.
Fourier integral.
Power spectrum.
Parseval's identity.
Operations.
Convolution theorem.
Transfer functions.
Correlation and Wiener-Khinchin theorem.
Discrete Fourier transform.
Numerical methods and FFT.
[Dispersion relations, Hilbert transforms, and Kramers-Kronig].
(15.1-15.7, [7.3])
Laplace Transforms
Other integral transforms.
Derivation from Fourier.
Properties.
Examples: ODEs, PDEs.
Periodic functions.
(15.8-15.12)
Probability
Combinatorics.
Probability distributions.
Binomial distribution.
Random walks.
Poisson distribution.
Poisson processes.
Gamma distributions.
Chi-squared distributions.
[Significance testing}].
Characteristic functions.
Central limit theorem.
(5.6, 10.5)
ODEs
First order ODEs (separable, exact, integrating factors, homogeneous).
Linear ODEs.
Linear independence and Wronskians.
Second solutions.
Particular integrals (undetermined coeeficients, variation of parameters,
Wronskian methods).
[Singular solutions].
Frobenius method.
Legendre polynomials.
Bessel functions.
Numerical methods.
(8.2, 8.4-8.6, 8.8)
Orthogonal functions
Examples.
Completeness.
Bessel's inequality.
Parseval's theorem.
Closure.
Orthogonalization.
[Linear vector spaces].
Sturm-Liouville theorem.
Hermitian operators.
Green's functions.
(9.1-9.4, 16.5)
Last updated: 28-Aug-95